Analysis and Approaches for IBDP Maths Ebook 2 | Page 192

Your Practice Set – Analysis and Approaches for IBDP Mathematics
Exercise 58
1. Consider the equation
log y �log x� log 16 2 .
3
8 2 8
(a)
Express x in terms of y , giving the answer in the form
x
b
� , where a , b� .
[7]
Let f be the function found in (a). R is the region bounded by the graph of f , the lines
y � 1 and y� c where c � 1, and the y -axis. It is given that the area of R is 2 2.
ay
(b) Show that c� e.
The function g is formed by reflecting f about the y -axis.
[5]
(c)
(d)
Show that
Hence, find
2 �
� ( f ( y) � g( y))dy
� 4 2 ln 2 , where � � 0 .
�
8000
� ( f ( y) � g( y))dy
.
125
[4]
[4]
2. The functions f and g are defined as
f x
2
( ) 4 x
� e and
g( x) � 4 x respectively, x� .
2
e �
(a) Find the coordinates of the point of intersection of the graphs of f and g .
R is the region bounded by the graph of f , the graph of g and the y -axis.
[5]
(b)
Show that the area of R is
2( e 1) ( e 1)
2 2
� � .
[7]
Let P( a, f ( a )) and Q( a, g( a )) be the points on the graphs of f and g respectively. It is
given that the rates of change of the gradient of the two graphs are equal.
(c) Show that a �2 � ln 4 .
(d) Hence, find the exact value of PQ .
[5]
[4]
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